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Geometric phase transferred from photonic mode to atomic BEC
Optics Communications 436 (2019) 52–56
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Geometric phase transferred from photonic mode to atomic BEC T.S. Yakovleva a,b , A.M. Rostom b , V.A. Tomilin a,b ,∗, L.V. Il’ichov a,b a b
Institute of Automation and Electrometry SB RAS, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia
ARTICLE
INFO
Keywords: Bose–Einstein condensate Two-mode approximation Geometric phase
ABSTRACT A process of geometric phase generation in a composite matter-field system is considered. Two atomic modes correspond to different localizations of a single Bose–Einstein condensate (BEC). One of the trapping localizations is formed by a photonic mode of a ring cavity. The photonic mode is governed by an external harmonic field source, by dissipation and by the number of localized atoms due to their non-resonant interaction with photons. This interaction gives rise to entanglement between the BEC and the photonic mode. By varying the intensity and frequency of the external source, it is possible to create a geometric phase for the optical mode. Because of the entanglement between the state of atomic and photonic modes, geometric phase acquired by the latter causes modification of the BEC state. This modification can be revealed by studying the tunneling between the atomic localizations.
1. Introduction
nonlinear system of equations. Another scheme was considered in [13], in which BEC was formed on a pair of hyperfine states with ‘internal’
Atomic Bose–Einstein condensate (BEC) is a macroscopic quantum system possessing spatial coherence. The latter manifests itself as interference of atoms from different spatial regions of the condensate. It is thus important to control phase relations between different regions of the condensate, as it is the principal unique feature of BEC that underlies its numerous technological and metrological prospects [1]. The technique of creating BEC localized in several minima of optical potential has recently received significant development [2–5]. Macroscopic coherence manifests itself as peculiarities of atomic tunneling between different localizations. Geometric phase can be used as an effective tool which modifies coherence properties between fractions of BEC. The concept of geometric phase was created by Pancharatnam [6] and Berry [7], its most general form was recently proposed in [8]. This is a purely kinematic entity defined by the trajectory of the evolving system’s state. Up to now it has been discovered in various systems belonging to quantum optics, molecular physics and condensed matter studies [9,10]. Up to our knowledge, investigations of geometric phase in BEC are currently at an early stage, with no experiments on creation and detection of geometric phase in atomic BEC. Theoretical proposals on the topic are also not numerous. In [11] a method for creating geometric Berry phase in BEC within a two-mode approximation (e.g., atoms occupying a pair of hyperfine states) was proposed, and several interesting consequences of atom–atom interaction were discovered. A similar model was discussed in [12]. The Berry phase was formed in a BEC localized in a double-well potential. The state of BEC was described in terms of a two-component wave function obeying a
Josephson junction created by Raman process. The authors managed to obtain analytical solution for BEC dynamics governed by a Hamiltonian with rather a general time dependence. In [14] the transformations in a system of paired BECs used for generation of geometric phase were shown to create entanglement between individual BECs. Aharonov– Bohm and Aharonov–Casher effects are intrinsically quite similar to geometric phase phenomenon. In [15] they studied the analogue of Aharonov–Casher effect in BEC localized in a toroidal magnetic trap. Also, a contribution of geometric phase appears in a Sagnac-effectdetection scheme based on matter-wave interferometry [16]. In optics, there exist elaborated methods of geometric phase generation and detection. Optical fields form dipole traps and can be used for localization and control of atomic BEC [17–21]. Control of atomic BEC’s state and dynamics is implemented by adjusting the parameters of electromagnetic fields. In most of experimental setups, the latter are easily controllable, especially when adiabatically slow variation of them is needed (which is usually the case in studies of geometric phase). Hence, it is of particular importance to seek for possible ways to create geometric phase in optical fields and then transfer it to atomic BEC. Entanglement as purely quantum correlations between the states of the optical trapping field and atoms can serve as a transfer channel. To illustrate this point, consider how an entangled state of BEC and photons
∗ Corresponding author at: Institute of Automation and Electrometry SB RAS, 630090 Novosibirsk, Russia. E-mail address: [email protected] (V.A. Tomilin).
https://doi.org/10.1016/j.optcom.2018.12.001 Received 17 September 2018; Received in revised form 25 November 2018; Accepted 1 December 2018 Available online 5 December 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.
T.S. Yakovleva, A.M. Rostom, V.A. Tomilin et al.
Optics Communications 436 (2019) 52–56
is modified by acquiring geometric phase: ∑ |𝛹 ⟩(𝑝𝑟𝑒) = 𝑓𝑖 |𝜑𝑖 ⟩𝑝ℎ ⊗ |𝜓𝑖 ⟩𝐵𝐸𝐶 ↦ ∑
𝑖
𝑓𝑖 𝑒𝚤𝜗𝑖 |𝜑𝑖 ⟩𝑝ℎ ⊗ |𝜓𝑖 ⟩𝐵𝐸𝐶 = |𝛹 ⟩(𝑝𝑜𝑠𝑡) .
(1)
𝑖
The BEC state 𝜌̂𝐵𝐸𝐶 = 𝑇 𝑟𝑝ℎ |𝛹 ⟩⟨𝛹 |, when considered separately, also undergoes transformation: ∑ ∑ 𝜌𝑖,𝑖′ 𝑒𝚤(𝜗𝑖 −𝜗𝑖′ ) |𝜓𝑖 ⟩⟨𝜓𝑖′ | = 𝜌̂(𝑝𝑜𝑠𝑡) . (2) 𝜌𝑖,𝑖′ |𝜓𝑖 ⟩⟨𝜓𝑖′ | ↦ 𝜌̂(𝑝𝑟𝑒) = 𝐵𝐸𝐶 𝐵𝐸𝐶 𝑖,𝑖′
𝑖,𝑖′
𝑓𝑖 𝑓𝑖∗′ ⟨𝜑𝑖′ |𝜑𝑖 ⟩.
Here 𝜌𝑖,𝑖′ = One can see that realization of this scheme requires two conditions: geometric phase must depend on the state of the photon subsystem (this usually demands no extra means) and the photonic states |𝜑𝑖 ⟩𝑝ℎ must be non-orthogonal. The aim of the present work is to consider a scheme that satisfies the aforementioned conditions. The article is organized as follows. In Section 2 we derive a master equation for the composite system of the light mode and the BEC relying on some previous results. We then implement approximation based on the assumed fast evolution of the light field, and in this limit obtain an evolution equation for the BEC itself. After that a correspondence between geometric phase acquired by the field and a certain modification of the BEC’s state is established. In Section 3 we introduce a toolkit suitable to analyze this modification in terms of a certain ‘phase’ parameter and present results of numerical calculations. Lastly, in Section 4 we summarize our results and suggest some prospects.
Fig. 1. Setup for observation of geometric phase in atomic BEC.
levels in the wells), as well as quadratic terms ∝ 𝜉𝑘 responsible for interaction between atoms in both localizations. Parameter 𝜒 gives the tunneling rate. The atomic mode 2 is assumed to be formed by the ring cavity optical beam with an average photon number 𝑛̄ 𝑝ℎ . Optical field is in non-resonant interaction with atoms forming the dipole trap, and the interaction strength is given by 𝑔. Parameters 𝜔2 , 𝜉2 and 𝜒 also depend on 𝑔 𝑛̄ 𝑝ℎ . The subtraction of this product from 𝜔2 is aimed to prevent double account of atoms-field interaction. The frequency 𝑔 represents the contribution of a single photon to optical trapping potential for mode 2. At the same time, it is also a shift of photonic mode’s eigenfrequency due to a single atom. The latter is reflected in the structure of the Liouvillian
2. The model We base our investigations on a scheme from [22,23]. Quantized optical mode in a ring cavity forms a minimum of optical potential. The mode is considered together with its harmonic external source and irreversible photon loss. The mode’s frequency is defined by the length of the cavity and also depends on the number of localized atoms. When changing this frequency, the equilibrium between the field source action and the loss of photons is shifted. This alters the steady-state of the mode and, consequently, the parameters of the trapping potential. Joint evolution of the subsystems (field and atoms) leads to a quasi-steady entangled state suitable for geometric phase transfer to atomic BEC. The consistent analysis of the system encounters significant mathematical problems, hence the model requires certain simplifications. The main simplifying assumption is about the evolution rate of the photon subsystem. Specifically, we shall assume it evolves much faster than the atomic subsystem that makes it adiabatically subordinate to the latter. In [23] a model with multiple atomic modes localized in a beam of a ring cavity was considered. In the context of the present work, a simpler model with a single oscillatory mode is more convenient. Optical potential has two minima: the first formed by an ordinary light beam and the second one formed by the quantized cavity mode, each creating a separate oscillatory mode for BEC (Fig. 1). It will be shown that the atomic tunneling between these minima is directly affected by the geometric phase. The theoretical model is essentially based on the master equation for the statistical operator 𝜚̂ of the compound system of the pair of atomic modes and the photonic mode of the cavity: 𝑑 (0) 𝜚̂ = −𝚤[𝐻̂ 𝑎𝑡 , 𝜚] ̂ + 𝛬𝑝ℎ [𝜚], ̂ 𝑑𝑡
𝛬𝑝ℎ [𝜚] ̂ = −𝚤𝛥[𝑎̂† 𝑎, ̂ 𝜚] ̂ − 𝚤𝑔[𝑎̂† 𝑎̂𝑛̂ 2 , 𝜚] ̂ + 𝛺[𝑎̂† − 𝑎, ̂ 𝜚]+ ̂ 𝛾(2𝑎̂𝜚̂𝑎̂† − 𝑎̂† 𝑎̂𝜚̂ − 𝜚̂𝑎̂† 𝑎). ̂
Here we have excluded the explicit time dependence of the classical field source with amplitude 𝛺; 𝛥 is its detuning from the eigenfrequency of the empty cavity; the second term represents the aforementioned frequency shift due to non-resonant interaction with atoms in the well 2. The last three terms describe irreducible photon loss from the cavity (𝛾 is the rate of this loss). It is once more worth noting that the term (𝜔2 − (0) 𝑔𝑛𝑝ℎ )𝑛̂ 2 in the Hamiltonian 𝐻̂ 𝑎𝑡 should be considered together with the second term in the Liouvillian. It provides a possibility to study quantum fluctuations of the photonic mode. These fluctuations should be small with respect to 𝑛̄ 𝑝ℎ for relevance of the implemented approximations. The equation governing the evolution of 𝑛̄ 𝑝ℎ was derived in [22,23]. As was mentioned above, the photonic subsystem is assumed to be fast. Its steady-state follows a slowly evolving state of the atomic modes. The total state of the three-mode system for a given number of atoms 𝑁 in BEC reads (see also [22,23])
It is written for two bosonic modes 𝑏̂ 1 (𝑏̂ †1 ) and 𝑏̂ 2
𝜚𝑛,𝑛′ |𝑁 − 𝑛⟩1 ⟨𝑁 − 𝑛′ | ⊗ |𝑛⟩2 ⟨𝑛′ | ⊗ |𝛼(𝑛)⟩⟨𝛼(𝑛′ )|.
(6)
𝑛,𝑛′ =0
Here 𝛼(𝑛) =
𝛺 . 𝚤(𝛥 + 𝑔𝑛) + 𝛾
(7)
The parts in (6) corresponding to atoms in the well 2 and the photonic mode appear to be eigenstates of the Liouvillian (5):
(3)
𝛬𝑝ℎ [|𝑛⟩2 ⟨𝑛′ | ⊗ |𝛼(𝑛)⟩⟨𝛼(𝑛′ )|] = 𝜆(𝑛, 𝑛′ )|𝑛⟩2 ⟨𝑛′ | ⊗ |𝛼(𝑛)⟩⟨𝛼(𝑛′ )|.
(8)
𝜆(𝑛, 𝑛′ )
Eigenvalues enter the right-hand side of the master equation for BEC and turn the condensate into an open system. It undergoes decoherence with respect to particle numbers in the atomic modes:
= 𝜔1 𝑛̂ 1 + (𝜔2 − 𝑔𝑛𝑝ℎ )𝑛̂ 2 +
𝜉1 𝑛̂ 1 (𝑛̂ 1 − 1) + 𝜉2 𝑛̂ 2 (𝑛̂ 2 − 1) + 𝜒(𝑏̂ †1 𝑏̂ 2 + 𝑏̂ †2 𝑏̂ 1 ).
𝑁 ∑
𝜚̂ =
(0) where atomic Hamiltonian 𝐻̂ 𝑎𝑡 and photonic Liouvillian 𝛬𝑝ℎ are introduced. The Hamiltonian has the form (0) 𝐻̂ 𝑎𝑡
(5)
(4) (𝑏̂ †2 ) – atoms in the wells
𝑑 𝜌̂ = −𝚤[𝐻̂ 𝑎𝑡 , 𝜌̂𝐵𝐸𝐶 ] + 2𝜈 𝑛̂ 2 𝜌̂𝐵𝐸𝐶 𝑛̂ 2 − 𝑑𝑡 𝐵𝐸𝐶 𝜈 𝑛̂ 22 𝜌̂𝐵𝐸𝐶 − 𝜈 𝜌̂𝐵𝐸𝐶 𝑛̂ 22 .
1 and 2, respectively – and contains the terms linear in the numbers of atoms 𝑛̂ 𝑘 = 𝑏̂ †𝑘 𝑏̂ 𝑘 (𝑘 = 1, 2; 𝜔𝑘 are the positions of ground oscillatory 53
(9)
T.S. Yakovleva, A.M. Rostom, V.A. Tomilin et al.
Optics Communications 436 (2019) 52–56
Here 𝑁 ∑
𝜌̂𝐵𝐸𝐶 =
𝜌𝑛 𝑛′ |𝑁 − 𝑛⟩1 ⟨𝑁 − 𝑛′ | ⊗ |𝑛⟩2 ⟨𝑛′ |,
(10)
𝑛,𝑛′ =0
𝜌𝑛,𝑛′ = 𝜚𝑛,𝑛′ ⟨𝛼(𝑛′ )|𝛼(𝑛)⟩,
(11)
𝜈 = 𝜆2 𝑔 2 𝛾∕(𝛥2 + 𝛾 2 )2 . (0) Dynamic-type terms from 𝜆(𝑛, 𝑛′ ) modify the Hamiltonian turning 𝐻̂ 𝑎𝑡 into 𝐻̂ 𝑎𝑡 [22,23]. Dissipative terms in (9) destroy the interference of different partitions of 𝑁 atoms between the pair of localizations. This is caused by leakage of information about the number of atoms in the mode 2. This information is encoded in phase properties of photons escaping the ring cavity. The off-diagonal matrix elements 𝜌𝑛,𝑛′ decay because of the dephasing. However, the rate 𝜈 of this process can be rather small. Realistic estimates in [22,23] give 𝜈 ∼ 10−2 s−1 . So, if the total duration of experiment is less than 𝜈 −1 , the phases of the off-diagonal matrix elements can effectively be controlled. Complex number (7) depends on two easily controllable parameters, 𝛺 and 𝛥. A closed curve in 𝛺𝛥-plane is transformed into some closed curve 𝑛 in the complex plane of 𝛼(𝑛) = 𝑥 + 𝚤𝑦, yielding the following geometric phase
𝜗𝑔 (𝑛) = 𝚤
∮𝑛
⟨𝛼|∇𝛼 |𝛼⟩𝑑𝛼 =
∮𝑛
(𝑦𝑑𝑥 − 𝑥𝑑𝑦).
(12)
The phase 𝜗𝑔 (𝑛) is given by oriented area (with coefficient −2) covered by the radius vector in 𝛼-plane when moving along the curve. That results in the following transformation of the off-diagonal matrix elements of 𝜌̂𝐵𝐸𝐶 : 𝜌(𝑝𝑟𝑒) ↦ 𝜌(𝑝𝑜𝑠𝑡) = 𝜌(𝑝𝑟𝑒) exp[𝚤𝜗𝑔 (𝑛) − 𝚤𝜗𝑔 (𝑛′ )]. 𝑛,𝑛′ 𝑛,𝑛′ 𝑛,𝑛′
Fig. 2. Contour of variation in the 𝛺𝛥-plane and in the 𝛼(𝑛)-plane for several values of 𝑔𝑛.
(13)
Naturally, this physical picture requires slow variation of 𝛺 and 𝛥 with respect to evolution rate of the photonic mode (this allows to stay within adiabaticity approximation). On the other hand, this evolution should be finished before the initial coherence between different atomic partitions is lost, i.e. the geometric phase should be generated rapidly with respect to the (rather long) lifetime of the off-diagonal matrix elements. The acquired geometric phase modifies the tunneling between the wells (in fact, it manifests itself through this modification). So, it is reasonable to interrupt the tunneling before the phase generation is completed. In general, BEC dynamics given by 𝐻̂ 𝑎𝑡 may also contribute to the final phase. The dynamical phase and its separation from the geometric phase is a problem discussed in Section 4.
The final expression of 𝑊 (𝜃, 𝑛) in terms of the matrix elements 𝜌𝑚,𝑛 of 𝜌̂ reads 1 𝑊 (𝜃, 𝑛) = 𝑇 𝑟[𝑉̂ (𝜃, 𝑛)† 𝜌] ̂ = 2𝜋 ) ( 𝑁 ∑ 1 −𝚤𝑚𝜃 𝚤𝑚𝜃 (𝑒 𝜌𝑚+𝑛,𝑛 + 𝑒 𝜌𝑛,𝑚+𝑛 ) . 𝜌𝑛,𝑛 + 2𝜋 𝑚=1
During that calculation, we have taken into account finite number of atoms in the BEC by setting 𝜌𝑛,𝑛′ = 0 for either of the two indices going beyond the interval [0, 𝑁]. The argument 𝑛 of the Wigner function is the number of atoms in atomic mode 2. However, the Wigner function defined that way characterizes the state of both atomic modes, since we used the matrix elements of the two-mode state (10). It is possible to construct a Wigner function of a single mode, by tracing over the basis of the other mode. However, this partial tracing eliminates all off-diagonal elements from the density matrix, making the resulting quantity insensitive to the geometric phase factors (13). Before calculating Wigner functions, one needs to specify the initial state of the BEC 𝜌̂(𝑝𝑟𝑒) and the curve in the space of 𝛺 and 𝛥 generating 𝐵𝐸𝐶 geometric phase 𝜗𝑔 . As 𝜌̂(𝑝𝑟𝑒) , we took the state with certain probabilities 𝐵𝐸𝐶 𝑝1 and 𝑝2 for an atom to occupy one well or another (𝑝1 + 𝑝2 = 1). The probability amplitudes 𝑓𝑛 from (11): √ 𝑛 𝑝𝑛 𝑝𝑁−𝑛 . 𝑓 𝑛 = 𝐶𝑁 (16) 1 2
3. Results Transformation of 𝜌̂(𝑝𝑟𝑒) into 𝜌̂(𝑝𝑜𝑠𝑡) given by (13) establishes relation between the matrix elements in the Fock basis. Relative phase of the BEC’s modes is encoded in the off-diagonal matrix elements. To demonstrate the effect of acquired geometric phase it is convenient to introduce an alternative representation of the quantum state as a function of the number of atoms 𝑛 in the BEC mode 2 and phase 𝜃. We take a Wigner-type function analogous to the known Wigner 𝑊 function for canonically conjugated position and momentum. There exist several variants of Wigner function for number and phase [24–26]. We choose the one proposed in [25] as the most consistent. It is based on a superoperator approach to a concept of phase in quantum mechanics. According to it, number-phase Wigner function 𝑊 (𝜃, 𝑛) of a state 𝜌̂ is defined as an expectation value of an eigenoperator 𝑉̂ (𝜃, 𝑛) of the phase superoperator: 𝑉̂ (𝜃, 𝑛) =
+∞ ∑
To simplify the calculations, the variation contour for 𝛺, 𝛥 was chosen to be rectangular (Fig. 2). Results of calculations of number-phase Wigner function for BEC state 𝜌̂(𝑝𝑟𝑒) prior to the acquisition of geometric phase are shown in 𝐵𝐸𝐶 Fig. 3(a). The arguments 𝑛 and 𝜃 are polar coordinates in the disk. The function has a single distinguished maximum at 𝑛 ≈ 𝑁 ⋅ 𝑝2 and 𝜃 near zero. Calculations of number-phase Wigner function for coherent state |𝛼⟩ show that it has a somewhat similar form with a maximum at 𝑛 ≈ |𝛼|2 and 𝜃 = 𝑎𝑟𝑔𝛼. Due to this, for comparison with the BEC Wigner function
𝑒𝚤𝑚𝜃 𝑉̂ (𝑚, 𝑛),
𝑚=−∞
𝑉̂ (𝑚, 𝑛) = ℎ(𝑚)|𝑚 + 𝑛⟩⟨𝑛| + ℎ(−1 − 𝑚)|𝑛⟩⟨𝑛 − 𝑚|,
(15)
(14)
where ℎ(𝑚) = 1 if 0 ≤ 𝑚 and ℎ(𝑚) = 0 if 𝑚 < 0, and 𝜃 varies from 0 to 2𝜋. 54
T.S. Yakovleva, A.M. Rostom, V.A. Tomilin et al.
Optics Communications 436 (2019) 52–56
√ a coherent state |𝛼 = 𝑁 ⋅ 𝑝2 ⟩ was chosen. It is plotted in Fig. 3(b) in the same circular region as the BEC Wigner functions. The resemblance is obvious, although the BEC Wigner function has a smaller uncertainty of the number variable. After variation through a contour listed in Fig. 2, the BEC appears in the state 𝜌̂(𝑝𝑜𝑠𝑡) . Its number-phase Wigner function 𝐵𝐸𝐶 is presented on Fig. 3(c). The most conspicuous change from that of the initial state 𝜌̂(𝑝𝑟𝑒) is its rotation through a certain angle (defined as 𝐵𝐸𝐶 a rotation of the function’s global maximum). Analysis of plots under various options shows that in the case of relatively weak atom-field interaction (𝑔 ≲ 𝛾, 𝛺) this angle only slightly depends on the initial state of the BEC (characterized by the wells’ occupation probabilities 𝑝1,2 ). Therefore, the geometric phase acquired by the optical field is capable to produce a significant change of the state of the BEC coupled to this field, and this change can be expressed in the form of a certain phase-like parameter. 4. Discussion In the Introduction we named two necessary conditions for geometric phase transfer from photonic system to BEC. Both conditions appear to be satisfied: geometric phase depends on the number of atoms in the photonic beam and the corresponding photonic states (Glauber coherent states) are non-orthogonal. The created geometric phase should reveal itself in interference of atoms from the localized modes. The simplest way to observe this interference is to study atomic tunneling between the modes. Due to the term ∝ 𝜒 in 𝐻̂ 𝑎𝑡 , a non-zero atomic flux appears: 𝐽̂ ≐ 𝚤[𝐻̂ 𝑎𝑡 , 𝑛̂ 1 − 𝑛̂ 2 ] = 2𝚤𝜒(𝑏̂ †2 𝑏̂ 1 − 𝑏̂ †1 𝑏̂ 2 ). In the initial state 𝜌̂(𝑝𝑟𝑒) 𝐵𝐸𝐶
(17)
defined in the previous section the average value
of the flux is ⟨𝐽̂⟩(𝑝𝑟𝑒) = 4𝜒
𝑁−1 ∑√
(𝑛 + 1)(𝑁 − 𝑛)𝜌𝑛,𝑛+1 ×
𝑛=0
𝑠𝑖𝑛(𝑎𝑟𝑔⟨𝛼(𝑛)|𝛼(𝑛 + 1)⟩).
(18)
while after acquiring geometric phase ⟨𝐽̂⟩(𝑝𝑜𝑠𝑡) = 4𝜒
𝑁−1 ∑√
(𝑛 + 1)(𝑁 − 𝑛)𝜌𝑛,𝑛+1 ×
𝑛=0
𝑠𝑖𝑛(𝜃𝑔 (𝑛 + 1) − 𝜃𝑔 (𝑛) + 𝑎𝑟𝑔⟨𝛼(𝑛)|𝛼(𝑛 + 1)⟩).
(19)
The flux changes drastically: for the rectangular contour presented on Fig. 2, upper part, and values of parameters listed in the caption to Fig. 3 ⟨𝐽̂⟩(𝑝𝑟𝑒) = −2.7𝜒, ⟨𝐽̂⟩(𝑝𝑜𝑠𝑡) = 17.6𝜒. Here we have only taken into account the geometric phase itself. In general, during its creation (which takes time 𝑇 ) the components of the state (10) may also acquire a phases due to BEC dynamics: 𝑇
𝜗𝑑 (𝑛) =
∫0
⟨𝑁 − 𝑛, 𝑛|𝐻̂ 𝑎𝑡 (𝑡)|𝑁 − 𝑛, 𝑛⟩𝑑𝑡.
(20) Fig. 3. Number-phase Wigner functions 𝑊 (𝜃, 𝑛) (𝑥 = 𝑛 ⋅ 𝑐𝑜𝑠𝜃, 𝑦 = 𝑛 ⋅ 𝑠𝑖𝑛𝜃) for (a) initial √ BEC state 𝜌̂(𝑝𝑟𝑒) with 𝑁 = 20, 𝑝1 = 0.3, 𝑔 = 𝛾∕10; (b) Glauber coherent state |𝛼 = 𝑁 ⋅ 𝑝2 ⟩; 𝐵𝐸𝐶
Since we assume the absence of tunneling during generation of geometric phase, |𝑁 − 𝑛⟩1 ⊗ |𝑛⟩2 are eigenstates of 𝐻̂ 𝑎𝑡 (𝑡) during the process. If interatomic forces in each mode may be neglected, there exists a way to eliminate the dynamical phase by a certain rearrangement of the studied system. Assume that mode 1 of the BEC is also formed in a waist of a ring cavity, similar to how the second mode is created. This modified system consists of two quantized photonic modes and two BEC modes. Let us excite the modes by a common source, dividing its energy in some variable proportion between them. If the initial point (identical to the final one) in the process of geometric phase generation is placed, for definiteness, in the middle of the right side of the rectangle (Fig. 2), cycling directions for each cavity are different. The emerging geometric phases of photonic modes are of the opposite signs, but do not in general compensate one other.1 The terms in the 1
(c) final BEC state 𝜌̂(𝑝𝑜𝑠𝑡) after the field parameters having been varied counterclockwise 𝐵𝐸𝐶 along a closed rectangular contour 1 ≤ 𝛺∕𝛾 ≤ 3, −1 ≤ 𝛥∕𝛾 ≤ 2.
dynamical phase when neglecting interatomic forces are proportional to the numbers of atoms in each localization. It can be easily seen that the resulting dynamical phase for the state |𝑁 − 𝑛⟩1 ⊗ |𝑛⟩2 no longer depends on 𝑛 and is proportional to 𝑁, making it unobservable. Switching on interaction between atoms breaks this physical picture. However, there is still a possibility to separate geometric phase from the dynamical phase. Geometric phase is independent on the speed of system’s traveling along the trajectory in 𝛺𝛥-plane, in contrast to dynamical phase. So, by controlling this speed, one can distinguish geometric phase from dynamical phase. Hence we have demonstrated the process of geometric phase creation in atomic BEC mediated by its entanglement with photonic mode.
Exact compensation only takes place if 𝑛 = 𝑁∕2. 55
T.S. Yakovleva, A.M. Rostom, V.A. Tomilin et al.
Optics Communications 436 (2019) 52–56
References
Actually, this phase is of optical nature and can be generated by varying the intensity and frequency of the external source exciting the cavity. There is a non-trivial generalization of the proposed scheme of lightto-matter transfer of geometric phase. In the present work we have considered the ‘linear’ field source. Its elementary interaction with the quantized mode is described in terms creation and annihilation of a single photon (see (5)). It was shown in [23] that the model can be extended to the case of pair-of-photons-type of source, when the cavity mode is excited in the process of parametric conversion. The steady photonic state |𝛼(𝑛)⟩⟨𝛼(𝑛)| for definite number of atoms is replaced by 𝜌̂𝑝ℎ (𝑛, 𝑛). This is a statistical operator of an effective thermal state deformed by a canonical transformation [23]. Now one must evaluate the operator 𝜌̂𝑝ℎ (𝑛, 𝑛′ ) which upon being taken in tensor product with |𝑛⟩2 ⟨𝑛′ | appears to be an eigenstate of the modified Liouvillian, similarly to (8): 𝛬𝑝ℎ [|𝑛⟩2 ⟨𝑛′ | ⊗ 𝜌̂𝑝ℎ (𝑛, 𝑛′ )|] = 𝜆(2) (𝑛, 𝑛′ )|𝑛⟩2 ⟨𝑛′ | ⊗ 𝜌̂𝑝ℎ (𝑛, 𝑛′ ).
[1] S. Martellucci, et al. (Eds.), Bose–Einstein Condensates and Atom Lasers, Kluwer Acad. Publ., New York, 2000. [2] M. Greiner, et al., Nature 415 (2002) 39. [3] J.H. Denschlag, et al., J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 3095. [4] K. Xu, et al., Phys. Rev. A 72 (2005) 043604. [5] T. Fukuhara, et al., Phys. Rev. A 79 (2009) 041604(R). [6] S. Pancharatnam, Proc. Indian Acad. Sci. Sect. A 44 (1956) 247. [7] M.V. Berry, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 392 (1984) 45. [8] N. Mukunda, R. Simon, Ann. Phys. 228 (1993) 205. [9] D. Chruśiński, A. Jamiolkovski, Geometric Phase in Classical and Quantum Mechanics, Birkhäuser, Boston, 2004. [10] A. Bohm, et al., The Geometric Phase in Quantum Systems, Springer-Verlag, Berlin, 2003. [11] I. Fuentes-Guridi, J. Pachos, S. Bose, V. Vedral, S. Choi, Phys. Rev. A 66 (2002) 022102. [12] R. Balakrishnan, M. Mehta, Eur. Phys. J. D 33 (2005) 437. [13] E.I. Duzzioni, L. Sanz, S.S. Mizrahi, M.H.Y. Moussa, Phys. Rev. A 75 (2007) 032113. [14] M.I. Hussain, E.O. Ilo-Okeke, T. Byrnes, Phys. Rev. A 89 (2014) 053607. [15] K.G. Petrosyan, L. You, Phys. Rev. A 59 (1999) 639. [16] O.I. Tolstikhin, T. Morishita, S. Watanabe, Phys. Rev. A 72 (2005) 051603(R). [17] T. Kinoshita, T. Wenger, D.S. Weiss, Phys. Rev. A 71 (2005) 011602(R). [18] R. Grimm, M. Weidemüller, Y.B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42 (2000) 95. [19] S.S. Szigeti, et al., Phys. Rev. A. 80 (2009) 013614. [20] C. Ottaviani, V. Ahufinger, R. Corbalán, J. Mompart, Phys. Rev. A 81 (2010) 043621. [21] A.C.J. Wade, J.F. Sherson, K. Mølmer, Phys. Rev. A 93 (2016) 023610. [22] L.V. Il’ichov, P.L. Chapovsky, Quantum Electron. 47 (5) (2017) 463. [23] L.V. Il’ichov, JETP Lett. 106 (1) (2017) 12. [24] W.K. Wooters, Ann. Phys. 176 (1987) 1. [25] M. Ban, Phys. Rev. A 51 (1995) 2469. [26] J.A. Vaccaro, Opt. Commun. 113 (1995) 421.
(21)
The same circular evolution in the 𝛺𝛥-plane must inevitably result in some geometric phase acquired by the off-diagonal operators 𝜌̂𝑝ℎ (𝑛, 𝑛′ ). This phase may have a more complicated form than a mere difference of two phases which depend on 𝑛 and 𝑛′ separately as in (13). This is not a trivial problem that deserves a further consideration. Acknowledgments The research was financially supported by the Ministry of Education and Science, Russia (project AAAA-A17-117052210003-4, the internal FASO number 0319-2016-0002).
56
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Geometric phase transferred from photonic mode to atomic BEC T.S. Yakovleva a,b , A.M. Rostom b , V.A. Tomilin a,b ,∗, L.V. Il’ichov a,b a b
Institute of Automation and Electrometry SB RAS, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia
ARTICLE
INFO
Keywords: Bose–Einstein condensate Two-mode approximation Geometric phase
ABSTRACT A process of geometric phase generation in a composite matter-field system is considered. Two atomic modes correspond to different localizations of a single Bose–Einstein condensate (BEC). One of the trapping localizations is formed by a photonic mode of a ring cavity. The photonic mode is governed by an external harmonic field source, by dissipation and by the number of localized atoms due to their non-resonant interaction with photons. This interaction gives rise to entanglement between the BEC and the photonic mode. By varying the intensity and frequency of the external source, it is possible to create a geometric phase for the optical mode. Because of the entanglement between the state of atomic and photonic modes, geometric phase acquired by the latter causes modification of the BEC state. This modification can be revealed by studying the tunneling between the atomic localizations.
1. Introduction
nonlinear system of equations. Another scheme was considered in [13], in which BEC was formed on a pair of hyperfine states with ‘internal’
Atomic Bose–Einstein condensate (BEC) is a macroscopic quantum system possessing spatial coherence. The latter manifests itself as interference of atoms from different spatial regions of the condensate. It is thus important to control phase relations between different regions of the condensate, as it is the principal unique feature of BEC that underlies its numerous technological and metrological prospects [1]. The technique of creating BEC localized in several minima of optical potential has recently received significant development [2–5]. Macroscopic coherence manifests itself as peculiarities of atomic tunneling between different localizations. Geometric phase can be used as an effective tool which modifies coherence properties between fractions of BEC. The concept of geometric phase was created by Pancharatnam [6] and Berry [7], its most general form was recently proposed in [8]. This is a purely kinematic entity defined by the trajectory of the evolving system’s state. Up to now it has been discovered in various systems belonging to quantum optics, molecular physics and condensed matter studies [9,10]. Up to our knowledge, investigations of geometric phase in BEC are currently at an early stage, with no experiments on creation and detection of geometric phase in atomic BEC. Theoretical proposals on the topic are also not numerous. In [11] a method for creating geometric Berry phase in BEC within a two-mode approximation (e.g., atoms occupying a pair of hyperfine states) was proposed, and several interesting consequences of atom–atom interaction were discovered. A similar model was discussed in [12]. The Berry phase was formed in a BEC localized in a double-well potential. The state of BEC was described in terms of a two-component wave function obeying a
Josephson junction created by Raman process. The authors managed to obtain analytical solution for BEC dynamics governed by a Hamiltonian with rather a general time dependence. In [14] the transformations in a system of paired BECs used for generation of geometric phase were shown to create entanglement between individual BECs. Aharonov– Bohm and Aharonov–Casher effects are intrinsically quite similar to geometric phase phenomenon. In [15] they studied the analogue of Aharonov–Casher effect in BEC localized in a toroidal magnetic trap. Also, a contribution of geometric phase appears in a Sagnac-effectdetection scheme based on matter-wave interferometry [16]. In optics, there exist elaborated methods of geometric phase generation and detection. Optical fields form dipole traps and can be used for localization and control of atomic BEC [17–21]. Control of atomic BEC’s state and dynamics is implemented by adjusting the parameters of electromagnetic fields. In most of experimental setups, the latter are easily controllable, especially when adiabatically slow variation of them is needed (which is usually the case in studies of geometric phase). Hence, it is of particular importance to seek for possible ways to create geometric phase in optical fields and then transfer it to atomic BEC. Entanglement as purely quantum correlations between the states of the optical trapping field and atoms can serve as a transfer channel. To illustrate this point, consider how an entangled state of BEC and photons
∗ Corresponding author at: Institute of Automation and Electrometry SB RAS, 630090 Novosibirsk, Russia. E-mail address: [email protected] (V.A. Tomilin).
https://doi.org/10.1016/j.optcom.2018.12.001 Received 17 September 2018; Received in revised form 25 November 2018; Accepted 1 December 2018 Available online 5 December 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.
T.S. Yakovleva, A.M. Rostom, V.A. Tomilin et al.
Optics Communications 436 (2019) 52–56
is modified by acquiring geometric phase: ∑ |𝛹 ⟩(𝑝𝑟𝑒) = 𝑓𝑖 |𝜑𝑖 ⟩𝑝ℎ ⊗ |𝜓𝑖 ⟩𝐵𝐸𝐶 ↦ ∑
𝑖
𝑓𝑖 𝑒𝚤𝜗𝑖 |𝜑𝑖 ⟩𝑝ℎ ⊗ |𝜓𝑖 ⟩𝐵𝐸𝐶 = |𝛹 ⟩(𝑝𝑜𝑠𝑡) .
(1)
𝑖
The BEC state 𝜌̂𝐵𝐸𝐶 = 𝑇 𝑟𝑝ℎ |𝛹 ⟩⟨𝛹 |, when considered separately, also undergoes transformation: ∑ ∑ 𝜌𝑖,𝑖′ 𝑒𝚤(𝜗𝑖 −𝜗𝑖′ ) |𝜓𝑖 ⟩⟨𝜓𝑖′ | = 𝜌̂(𝑝𝑜𝑠𝑡) . (2) 𝜌𝑖,𝑖′ |𝜓𝑖 ⟩⟨𝜓𝑖′ | ↦ 𝜌̂(𝑝𝑟𝑒) = 𝐵𝐸𝐶 𝐵𝐸𝐶 𝑖,𝑖′
𝑖,𝑖′
𝑓𝑖 𝑓𝑖∗′ ⟨𝜑𝑖′ |𝜑𝑖 ⟩.
Here 𝜌𝑖,𝑖′ = One can see that realization of this scheme requires two conditions: geometric phase must depend on the state of the photon subsystem (this usually demands no extra means) and the photonic states |𝜑𝑖 ⟩𝑝ℎ must be non-orthogonal. The aim of the present work is to consider a scheme that satisfies the aforementioned conditions. The article is organized as follows. In Section 2 we derive a master equation for the composite system of the light mode and the BEC relying on some previous results. We then implement approximation based on the assumed fast evolution of the light field, and in this limit obtain an evolution equation for the BEC itself. After that a correspondence between geometric phase acquired by the field and a certain modification of the BEC’s state is established. In Section 3 we introduce a toolkit suitable to analyze this modification in terms of a certain ‘phase’ parameter and present results of numerical calculations. Lastly, in Section 4 we summarize our results and suggest some prospects.
Fig. 1. Setup for observation of geometric phase in atomic BEC.
levels in the wells), as well as quadratic terms ∝ 𝜉𝑘 responsible for interaction between atoms in both localizations. Parameter 𝜒 gives the tunneling rate. The atomic mode 2 is assumed to be formed by the ring cavity optical beam with an average photon number 𝑛̄ 𝑝ℎ . Optical field is in non-resonant interaction with atoms forming the dipole trap, and the interaction strength is given by 𝑔. Parameters 𝜔2 , 𝜉2 and 𝜒 also depend on 𝑔 𝑛̄ 𝑝ℎ . The subtraction of this product from 𝜔2 is aimed to prevent double account of atoms-field interaction. The frequency 𝑔 represents the contribution of a single photon to optical trapping potential for mode 2. At the same time, it is also a shift of photonic mode’s eigenfrequency due to a single atom. The latter is reflected in the structure of the Liouvillian
2. The model We base our investigations on a scheme from [22,23]. Quantized optical mode in a ring cavity forms a minimum of optical potential. The mode is considered together with its harmonic external source and irreversible photon loss. The mode’s frequency is defined by the length of the cavity and also depends on the number of localized atoms. When changing this frequency, the equilibrium between the field source action and the loss of photons is shifted. This alters the steady-state of the mode and, consequently, the parameters of the trapping potential. Joint evolution of the subsystems (field and atoms) leads to a quasi-steady entangled state suitable for geometric phase transfer to atomic BEC. The consistent analysis of the system encounters significant mathematical problems, hence the model requires certain simplifications. The main simplifying assumption is about the evolution rate of the photon subsystem. Specifically, we shall assume it evolves much faster than the atomic subsystem that makes it adiabatically subordinate to the latter. In [23] a model with multiple atomic modes localized in a beam of a ring cavity was considered. In the context of the present work, a simpler model with a single oscillatory mode is more convenient. Optical potential has two minima: the first formed by an ordinary light beam and the second one formed by the quantized cavity mode, each creating a separate oscillatory mode for BEC (Fig. 1). It will be shown that the atomic tunneling between these minima is directly affected by the geometric phase. The theoretical model is essentially based on the master equation for the statistical operator 𝜚̂ of the compound system of the pair of atomic modes and the photonic mode of the cavity: 𝑑 (0) 𝜚̂ = −𝚤[𝐻̂ 𝑎𝑡 , 𝜚] ̂ + 𝛬𝑝ℎ [𝜚], ̂ 𝑑𝑡
𝛬𝑝ℎ [𝜚] ̂ = −𝚤𝛥[𝑎̂† 𝑎, ̂ 𝜚] ̂ − 𝚤𝑔[𝑎̂† 𝑎̂𝑛̂ 2 , 𝜚] ̂ + 𝛺[𝑎̂† − 𝑎, ̂ 𝜚]+ ̂ 𝛾(2𝑎̂𝜚̂𝑎̂† − 𝑎̂† 𝑎̂𝜚̂ − 𝜚̂𝑎̂† 𝑎). ̂
Here we have excluded the explicit time dependence of the classical field source with amplitude 𝛺; 𝛥 is its detuning from the eigenfrequency of the empty cavity; the second term represents the aforementioned frequency shift due to non-resonant interaction with atoms in the well 2. The last three terms describe irreducible photon loss from the cavity (𝛾 is the rate of this loss). It is once more worth noting that the term (𝜔2 − (0) 𝑔𝑛𝑝ℎ )𝑛̂ 2 in the Hamiltonian 𝐻̂ 𝑎𝑡 should be considered together with the second term in the Liouvillian. It provides a possibility to study quantum fluctuations of the photonic mode. These fluctuations should be small with respect to 𝑛̄ 𝑝ℎ for relevance of the implemented approximations. The equation governing the evolution of 𝑛̄ 𝑝ℎ was derived in [22,23]. As was mentioned above, the photonic subsystem is assumed to be fast. Its steady-state follows a slowly evolving state of the atomic modes. The total state of the three-mode system for a given number of atoms 𝑁 in BEC reads (see also [22,23])
It is written for two bosonic modes 𝑏̂ 1 (𝑏̂ †1 ) and 𝑏̂ 2
𝜚𝑛,𝑛′ |𝑁 − 𝑛⟩1 ⟨𝑁 − 𝑛′ | ⊗ |𝑛⟩2 ⟨𝑛′ | ⊗ |𝛼(𝑛)⟩⟨𝛼(𝑛′ )|.
(6)
𝑛,𝑛′ =0
Here 𝛼(𝑛) =
𝛺 . 𝚤(𝛥 + 𝑔𝑛) + 𝛾
(7)
The parts in (6) corresponding to atoms in the well 2 and the photonic mode appear to be eigenstates of the Liouvillian (5):
(3)
𝛬𝑝ℎ [|𝑛⟩2 ⟨𝑛′ | ⊗ |𝛼(𝑛)⟩⟨𝛼(𝑛′ )|] = 𝜆(𝑛, 𝑛′ )|𝑛⟩2 ⟨𝑛′ | ⊗ |𝛼(𝑛)⟩⟨𝛼(𝑛′ )|.
(8)
𝜆(𝑛, 𝑛′ )
Eigenvalues enter the right-hand side of the master equation for BEC and turn the condensate into an open system. It undergoes decoherence with respect to particle numbers in the atomic modes:
= 𝜔1 𝑛̂ 1 + (𝜔2 − 𝑔𝑛𝑝ℎ )𝑛̂ 2 +
𝜉1 𝑛̂ 1 (𝑛̂ 1 − 1) + 𝜉2 𝑛̂ 2 (𝑛̂ 2 − 1) + 𝜒(𝑏̂ †1 𝑏̂ 2 + 𝑏̂ †2 𝑏̂ 1 ).
𝑁 ∑
𝜚̂ =
(0) where atomic Hamiltonian 𝐻̂ 𝑎𝑡 and photonic Liouvillian 𝛬𝑝ℎ are introduced. The Hamiltonian has the form (0) 𝐻̂ 𝑎𝑡
(5)
(4) (𝑏̂ †2 ) – atoms in the wells
𝑑 𝜌̂ = −𝚤[𝐻̂ 𝑎𝑡 , 𝜌̂𝐵𝐸𝐶 ] + 2𝜈 𝑛̂ 2 𝜌̂𝐵𝐸𝐶 𝑛̂ 2 − 𝑑𝑡 𝐵𝐸𝐶 𝜈 𝑛̂ 22 𝜌̂𝐵𝐸𝐶 − 𝜈 𝜌̂𝐵𝐸𝐶 𝑛̂ 22 .
1 and 2, respectively – and contains the terms linear in the numbers of atoms 𝑛̂ 𝑘 = 𝑏̂ †𝑘 𝑏̂ 𝑘 (𝑘 = 1, 2; 𝜔𝑘 are the positions of ground oscillatory 53
(9)
T.S. Yakovleva, A.M. Rostom, V.A. Tomilin et al.
Optics Communications 436 (2019) 52–56
Here 𝑁 ∑
𝜌̂𝐵𝐸𝐶 =
𝜌𝑛 𝑛′ |𝑁 − 𝑛⟩1 ⟨𝑁 − 𝑛′ | ⊗ |𝑛⟩2 ⟨𝑛′ |,
(10)
𝑛,𝑛′ =0
𝜌𝑛,𝑛′ = 𝜚𝑛,𝑛′ ⟨𝛼(𝑛′ )|𝛼(𝑛)⟩,
(11)
𝜈 = 𝜆2 𝑔 2 𝛾∕(𝛥2 + 𝛾 2 )2 . (0) Dynamic-type terms from 𝜆(𝑛, 𝑛′ ) modify the Hamiltonian turning 𝐻̂ 𝑎𝑡 into 𝐻̂ 𝑎𝑡 [22,23]. Dissipative terms in (9) destroy the interference of different partitions of 𝑁 atoms between the pair of localizations. This is caused by leakage of information about the number of atoms in the mode 2. This information is encoded in phase properties of photons escaping the ring cavity. The off-diagonal matrix elements 𝜌𝑛,𝑛′ decay because of the dephasing. However, the rate 𝜈 of this process can be rather small. Realistic estimates in [22,23] give 𝜈 ∼ 10−2 s−1 . So, if the total duration of experiment is less than 𝜈 −1 , the phases of the off-diagonal matrix elements can effectively be controlled. Complex number (7) depends on two easily controllable parameters, 𝛺 and 𝛥. A closed curve in 𝛺𝛥-plane is transformed into some closed curve 𝑛 in the complex plane of 𝛼(𝑛) = 𝑥 + 𝚤𝑦, yielding the following geometric phase
𝜗𝑔 (𝑛) = 𝚤
∮𝑛
⟨𝛼|∇𝛼 |𝛼⟩𝑑𝛼 =
∮𝑛
(𝑦𝑑𝑥 − 𝑥𝑑𝑦).
(12)
The phase 𝜗𝑔 (𝑛) is given by oriented area (with coefficient −2) covered by the radius vector in 𝛼-plane when moving along the curve. That results in the following transformation of the off-diagonal matrix elements of 𝜌̂𝐵𝐸𝐶 : 𝜌(𝑝𝑟𝑒) ↦ 𝜌(𝑝𝑜𝑠𝑡) = 𝜌(𝑝𝑟𝑒) exp[𝚤𝜗𝑔 (𝑛) − 𝚤𝜗𝑔 (𝑛′ )]. 𝑛,𝑛′ 𝑛,𝑛′ 𝑛,𝑛′
Fig. 2. Contour of variation in the 𝛺𝛥-plane and in the 𝛼(𝑛)-plane for several values of 𝑔𝑛.
(13)
Naturally, this physical picture requires slow variation of 𝛺 and 𝛥 with respect to evolution rate of the photonic mode (this allows to stay within adiabaticity approximation). On the other hand, this evolution should be finished before the initial coherence between different atomic partitions is lost, i.e. the geometric phase should be generated rapidly with respect to the (rather long) lifetime of the off-diagonal matrix elements. The acquired geometric phase modifies the tunneling between the wells (in fact, it manifests itself through this modification). So, it is reasonable to interrupt the tunneling before the phase generation is completed. In general, BEC dynamics given by 𝐻̂ 𝑎𝑡 may also contribute to the final phase. The dynamical phase and its separation from the geometric phase is a problem discussed in Section 4.
The final expression of 𝑊 (𝜃, 𝑛) in terms of the matrix elements 𝜌𝑚,𝑛 of 𝜌̂ reads 1 𝑊 (𝜃, 𝑛) = 𝑇 𝑟[𝑉̂ (𝜃, 𝑛)† 𝜌] ̂ = 2𝜋 ) ( 𝑁 ∑ 1 −𝚤𝑚𝜃 𝚤𝑚𝜃 (𝑒 𝜌𝑚+𝑛,𝑛 + 𝑒 𝜌𝑛,𝑚+𝑛 ) . 𝜌𝑛,𝑛 + 2𝜋 𝑚=1
During that calculation, we have taken into account finite number of atoms in the BEC by setting 𝜌𝑛,𝑛′ = 0 for either of the two indices going beyond the interval [0, 𝑁]. The argument 𝑛 of the Wigner function is the number of atoms in atomic mode 2. However, the Wigner function defined that way characterizes the state of both atomic modes, since we used the matrix elements of the two-mode state (10). It is possible to construct a Wigner function of a single mode, by tracing over the basis of the other mode. However, this partial tracing eliminates all off-diagonal elements from the density matrix, making the resulting quantity insensitive to the geometric phase factors (13). Before calculating Wigner functions, one needs to specify the initial state of the BEC 𝜌̂(𝑝𝑟𝑒) and the curve in the space of 𝛺 and 𝛥 generating 𝐵𝐸𝐶 geometric phase 𝜗𝑔 . As 𝜌̂(𝑝𝑟𝑒) , we took the state with certain probabilities 𝐵𝐸𝐶 𝑝1 and 𝑝2 for an atom to occupy one well or another (𝑝1 + 𝑝2 = 1). The probability amplitudes 𝑓𝑛 from (11): √ 𝑛 𝑝𝑛 𝑝𝑁−𝑛 . 𝑓 𝑛 = 𝐶𝑁 (16) 1 2
3. Results Transformation of 𝜌̂(𝑝𝑟𝑒) into 𝜌̂(𝑝𝑜𝑠𝑡) given by (13) establishes relation between the matrix elements in the Fock basis. Relative phase of the BEC’s modes is encoded in the off-diagonal matrix elements. To demonstrate the effect of acquired geometric phase it is convenient to introduce an alternative representation of the quantum state as a function of the number of atoms 𝑛 in the BEC mode 2 and phase 𝜃. We take a Wigner-type function analogous to the known Wigner 𝑊 function for canonically conjugated position and momentum. There exist several variants of Wigner function for number and phase [24–26]. We choose the one proposed in [25] as the most consistent. It is based on a superoperator approach to a concept of phase in quantum mechanics. According to it, number-phase Wigner function 𝑊 (𝜃, 𝑛) of a state 𝜌̂ is defined as an expectation value of an eigenoperator 𝑉̂ (𝜃, 𝑛) of the phase superoperator: 𝑉̂ (𝜃, 𝑛) =
+∞ ∑
To simplify the calculations, the variation contour for 𝛺, 𝛥 was chosen to be rectangular (Fig. 2). Results of calculations of number-phase Wigner function for BEC state 𝜌̂(𝑝𝑟𝑒) prior to the acquisition of geometric phase are shown in 𝐵𝐸𝐶 Fig. 3(a). The arguments 𝑛 and 𝜃 are polar coordinates in the disk. The function has a single distinguished maximum at 𝑛 ≈ 𝑁 ⋅ 𝑝2 and 𝜃 near zero. Calculations of number-phase Wigner function for coherent state |𝛼⟩ show that it has a somewhat similar form with a maximum at 𝑛 ≈ |𝛼|2 and 𝜃 = 𝑎𝑟𝑔𝛼. Due to this, for comparison with the BEC Wigner function
𝑒𝚤𝑚𝜃 𝑉̂ (𝑚, 𝑛),
𝑚=−∞
𝑉̂ (𝑚, 𝑛) = ℎ(𝑚)|𝑚 + 𝑛⟩⟨𝑛| + ℎ(−1 − 𝑚)|𝑛⟩⟨𝑛 − 𝑚|,
(15)
(14)
where ℎ(𝑚) = 1 if 0 ≤ 𝑚 and ℎ(𝑚) = 0 if 𝑚 < 0, and 𝜃 varies from 0 to 2𝜋. 54
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Optics Communications 436 (2019) 52–56
√ a coherent state |𝛼 = 𝑁 ⋅ 𝑝2 ⟩ was chosen. It is plotted in Fig. 3(b) in the same circular region as the BEC Wigner functions. The resemblance is obvious, although the BEC Wigner function has a smaller uncertainty of the number variable. After variation through a contour listed in Fig. 2, the BEC appears in the state 𝜌̂(𝑝𝑜𝑠𝑡) . Its number-phase Wigner function 𝐵𝐸𝐶 is presented on Fig. 3(c). The most conspicuous change from that of the initial state 𝜌̂(𝑝𝑟𝑒) is its rotation through a certain angle (defined as 𝐵𝐸𝐶 a rotation of the function’s global maximum). Analysis of plots under various options shows that in the case of relatively weak atom-field interaction (𝑔 ≲ 𝛾, 𝛺) this angle only slightly depends on the initial state of the BEC (characterized by the wells’ occupation probabilities 𝑝1,2 ). Therefore, the geometric phase acquired by the optical field is capable to produce a significant change of the state of the BEC coupled to this field, and this change can be expressed in the form of a certain phase-like parameter. 4. Discussion In the Introduction we named two necessary conditions for geometric phase transfer from photonic system to BEC. Both conditions appear to be satisfied: geometric phase depends on the number of atoms in the photonic beam and the corresponding photonic states (Glauber coherent states) are non-orthogonal. The created geometric phase should reveal itself in interference of atoms from the localized modes. The simplest way to observe this interference is to study atomic tunneling between the modes. Due to the term ∝ 𝜒 in 𝐻̂ 𝑎𝑡 , a non-zero atomic flux appears: 𝐽̂ ≐ 𝚤[𝐻̂ 𝑎𝑡 , 𝑛̂ 1 − 𝑛̂ 2 ] = 2𝚤𝜒(𝑏̂ †2 𝑏̂ 1 − 𝑏̂ †1 𝑏̂ 2 ). In the initial state 𝜌̂(𝑝𝑟𝑒) 𝐵𝐸𝐶
(17)
defined in the previous section the average value
of the flux is ⟨𝐽̂⟩(𝑝𝑟𝑒) = 4𝜒
𝑁−1 ∑√
(𝑛 + 1)(𝑁 − 𝑛)𝜌𝑛,𝑛+1 ×
𝑛=0
𝑠𝑖𝑛(𝑎𝑟𝑔⟨𝛼(𝑛)|𝛼(𝑛 + 1)⟩).
(18)
while after acquiring geometric phase ⟨𝐽̂⟩(𝑝𝑜𝑠𝑡) = 4𝜒
𝑁−1 ∑√
(𝑛 + 1)(𝑁 − 𝑛)𝜌𝑛,𝑛+1 ×
𝑛=0
𝑠𝑖𝑛(𝜃𝑔 (𝑛 + 1) − 𝜃𝑔 (𝑛) + 𝑎𝑟𝑔⟨𝛼(𝑛)|𝛼(𝑛 + 1)⟩).
(19)
The flux changes drastically: for the rectangular contour presented on Fig. 2, upper part, and values of parameters listed in the caption to Fig. 3 ⟨𝐽̂⟩(𝑝𝑟𝑒) = −2.7𝜒, ⟨𝐽̂⟩(𝑝𝑜𝑠𝑡) = 17.6𝜒. Here we have only taken into account the geometric phase itself. In general, during its creation (which takes time 𝑇 ) the components of the state (10) may also acquire a phases due to BEC dynamics: 𝑇
𝜗𝑑 (𝑛) =
∫0
⟨𝑁 − 𝑛, 𝑛|𝐻̂ 𝑎𝑡 (𝑡)|𝑁 − 𝑛, 𝑛⟩𝑑𝑡.
(20) Fig. 3. Number-phase Wigner functions 𝑊 (𝜃, 𝑛) (𝑥 = 𝑛 ⋅ 𝑐𝑜𝑠𝜃, 𝑦 = 𝑛 ⋅ 𝑠𝑖𝑛𝜃) for (a) initial √ BEC state 𝜌̂(𝑝𝑟𝑒) with 𝑁 = 20, 𝑝1 = 0.3, 𝑔 = 𝛾∕10; (b) Glauber coherent state |𝛼 = 𝑁 ⋅ 𝑝2 ⟩; 𝐵𝐸𝐶
Since we assume the absence of tunneling during generation of geometric phase, |𝑁 − 𝑛⟩1 ⊗ |𝑛⟩2 are eigenstates of 𝐻̂ 𝑎𝑡 (𝑡) during the process. If interatomic forces in each mode may be neglected, there exists a way to eliminate the dynamical phase by a certain rearrangement of the studied system. Assume that mode 1 of the BEC is also formed in a waist of a ring cavity, similar to how the second mode is created. This modified system consists of two quantized photonic modes and two BEC modes. Let us excite the modes by a common source, dividing its energy in some variable proportion between them. If the initial point (identical to the final one) in the process of geometric phase generation is placed, for definiteness, in the middle of the right side of the rectangle (Fig. 2), cycling directions for each cavity are different. The emerging geometric phases of photonic modes are of the opposite signs, but do not in general compensate one other.1 The terms in the 1
(c) final BEC state 𝜌̂(𝑝𝑜𝑠𝑡) after the field parameters having been varied counterclockwise 𝐵𝐸𝐶 along a closed rectangular contour 1 ≤ 𝛺∕𝛾 ≤ 3, −1 ≤ 𝛥∕𝛾 ≤ 2.
dynamical phase when neglecting interatomic forces are proportional to the numbers of atoms in each localization. It can be easily seen that the resulting dynamical phase for the state |𝑁 − 𝑛⟩1 ⊗ |𝑛⟩2 no longer depends on 𝑛 and is proportional to 𝑁, making it unobservable. Switching on interaction between atoms breaks this physical picture. However, there is still a possibility to separate geometric phase from the dynamical phase. Geometric phase is independent on the speed of system’s traveling along the trajectory in 𝛺𝛥-plane, in contrast to dynamical phase. So, by controlling this speed, one can distinguish geometric phase from dynamical phase. Hence we have demonstrated the process of geometric phase creation in atomic BEC mediated by its entanglement with photonic mode.
Exact compensation only takes place if 𝑛 = 𝑁∕2. 55
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Optics Communications 436 (2019) 52–56
References
Actually, this phase is of optical nature and can be generated by varying the intensity and frequency of the external source exciting the cavity. There is a non-trivial generalization of the proposed scheme of lightto-matter transfer of geometric phase. In the present work we have considered the ‘linear’ field source. Its elementary interaction with the quantized mode is described in terms creation and annihilation of a single photon (see (5)). It was shown in [23] that the model can be extended to the case of pair-of-photons-type of source, when the cavity mode is excited in the process of parametric conversion. The steady photonic state |𝛼(𝑛)⟩⟨𝛼(𝑛)| for definite number of atoms is replaced by 𝜌̂𝑝ℎ (𝑛, 𝑛). This is a statistical operator of an effective thermal state deformed by a canonical transformation [23]. Now one must evaluate the operator 𝜌̂𝑝ℎ (𝑛, 𝑛′ ) which upon being taken in tensor product with |𝑛⟩2 ⟨𝑛′ | appears to be an eigenstate of the modified Liouvillian, similarly to (8): 𝛬𝑝ℎ [|𝑛⟩2 ⟨𝑛′ | ⊗ 𝜌̂𝑝ℎ (𝑛, 𝑛′ )|] = 𝜆(2) (𝑛, 𝑛′ )|𝑛⟩2 ⟨𝑛′ | ⊗ 𝜌̂𝑝ℎ (𝑛, 𝑛′ ).
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(21)
The same circular evolution in the 𝛺𝛥-plane must inevitably result in some geometric phase acquired by the off-diagonal operators 𝜌̂𝑝ℎ (𝑛, 𝑛′ ). This phase may have a more complicated form than a mere difference of two phases which depend on 𝑛 and 𝑛′ separately as in (13). This is not a trivial problem that deserves a further consideration. Acknowledgments The research was financially supported by the Ministry of Education and Science, Russia (project AAAA-A17-117052210003-4, the internal FASO number 0319-2016-0002).
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